An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i.e., non-permanently) when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region:
  λ  ⁢      =    def    ⁢      stress    strain  where lambda (λ) is the elastic modulus; stress is the force causing the deformation divided by the area to which the force is applied; and strain is the ratio of the change caused by the stress to the original state of the object. If stress is measured in Pascal, since strain is a unitless ratio, then the units of λ are Pascal as well. Since the denominator becomes unity if length is doubled, the elastic modulus becomes the stress needed to cause a sample of the material to double in length. While this endpoint is not realistic because most materials will fail before reaching it, it is practical, in that small fractions of the defining load will operate in exactly the same ratio. Thus for steel with an elastic modulus of 30 million pounds per square inch, a 30 thousand psi load will elongate a 1 inch bar by one thousandth of an inch, and similarly for metric units, where a thousandth of the modulus in GPascal (GPa) will change a meter by a millimeter.
Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are: Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus. In the present application elastic modulus primarily relates to the Young's modulus; the shear modulus or modulus of rigidity (G or μ) describes an object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain. The shear modulus is part of the derivation of viscosity; and the bulk modulus (K) describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young's modulus to three dimensions.
The ITRS roadmap for scaling of ultra-large-scale integrated circuits requires mechanically robust dielectric materials with a low k-value. Low-k materials currently used in Cu/low-k integration scheme have k-values between 2.7 and 3.0. One of the limiting factors in further reduction of k-value is mechanical robustness, since more than 32% of porosity needs to be introduced to a Plasma Enhanced Chemically Vapor Deposited (PE-CVD) or Chemically Vapor deposited (CVD) low-k film to achieve the k-values below 2.3.
In state of the art (Kemeling et al. in Microelectronic Engineering Volume 84, Issue 11, November 2007, Pages 2575-2581) PE-CVD deposited low-k films such as Aurora® ELK films are fabricated by PE-CVD of a SiCOH matrix precursor and an organic porogen material. The porogen material is then removed during a subsequent thermally assisted UV-cure step with a short wavelength UV-lamp (λ<200 nm). In the best case this results in film thickness shrinkage of 13.2% and a robust low-k film with k-value of 2.3 and elastic modulus of 5.0 GPa.
A further increase in elastic modulus without altering the k value (porosity) and/or the chemical stability is desired in order to withstand further processing and reliability of the device (such as dry-etch patterning or chemical mechanical polishing (CMP) process).